13th International Conference on Fracture June 16–21, 2013, Beijing, China -4- growth of the periodic edge cracks, however, is caused by the longitudinal stress which has the following form when the cracks have not been considered ( ) ( ) ( ) 2 0 0 , 1 1 ( , ) 4 6 , 6 12 , 1 1 b b T yy E x E x x x x dx x x dx b b b b αθ τ σ τ αθ τ α θ τ ν ν ⎡ ⎤ ⎞ ⎞ ⎛ ⎛ =− + − − − ⎢ ⎥ ⎜ ⎟ ⎜ ⎟ − − ⎝ ⎝ ⎠ ⎠ ⎣ ⎦ ∫ ∫ , (9) where θ(x, τ) = T(x, τ)-T0, E is Young’s modulus, ν is Poisson’s ratio, and α = α(x) is the coefficient of thermal expansion. Here we assume that the FGM plate is thermally nonhomogeneous but elastically homogeneous, i.e., the Young’s modulus and Poisson’s ratio are constant. While this assumption imposes limitations on the application of the present model, there exist some FGM systems for which the Young's modulus remains approximately constant. Examples include TiC/SiC, MoSi2/Al2O3, Al2O3/Si3N4, and ZrO2/Nickel FGM systems. 3. Thermal Stress Intensity Factors This section uses the asymptotic temperature solution (4) – (6) to calculate the thermal stress intensity factor (TSIFs) at the tips of long and short crack in an elastically homogeneous but thermally graded FGM plate (see Fig. 1). The integral equation method is employed and the singular integral equations of the crack problem are given as follows 1 1 1 2 11 1 1 1 1 1 12 2 2 2 2 2 1 1 1 1 2 1 1 1 ( , ) ( ) ( , ) ( ) 2 2 2 (1 ) ( , ), 1 1, T yy a a k r s f s ds k r s f s ds s r r t r E π ν σ − − ⎡ ⎤ + + ⎢ ⎥ ⎣ − ⎦ − =− − ≤ ≤ ∫ ∫ (10a) 1 1 2 1 22 2 2 2 2 2 21 1 1 1 1 1 1 2 2 1 2 2 2 1 ( , ) ( ) ( , ) ( ) 2 2 2 (1 ) ( , ), 1 1, T yy a a k r s f s ds k r s f s ds s r r t r E π ν σ − − ⎡ ⎤ + + ⎢ ⎥ ⎣ − ⎦ − =− − ≤ ≤ ∫ ∫ (10b) where the basic unknown variables f1(x1) and f2(x2) are defined by (1) (2) 1 2 1 1 2 2 1 2 ( ,0 ) ( ,0 ) ( ) , ( ) v x v x f x f x x x + + ∂ ∂ = = ∂ ∂ , (11) with v(1)(x 1, y1) and v (2)(x 2, y2) being the displacements in the y-direction, x1, y1, x2 and y2 are coordinates defined by 1 1 2 2 , , , . x x y y x x y y h = = = = − (12) r1, s1, r2, and s2 are the normalized coordinates 1 1 1 1 1 1 2 2 2 2 2 2 (1 ), (1 ), 2 2 (1 ), (1 ). 2 2 a a x r x s a a x r x s ′ = + = + ′ = + = + (13) k11, k12, k21 and k22 are known kernels given by Feng and Jin[10], and T yyσ is given in Eq. (9).
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